Theorem spw 2038

Description: Weak version of the specialization scheme sp 2178. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2178 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2178 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2133 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2178 are spfw 2037 (minimal distinct variable requirements), spnfw 1984 (when 𝑥 is not free in ¬ 𝜑), spvw 1985 (when 𝑥 does not appear in 𝜑), sptruw 1810 (when 𝜑 is true), spfalw 2002 (when 𝜑 is false), and spvv 2001 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1914	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1914	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1914	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2037	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

This theorem is referenced by:  hba1w  2051  19.8aw  2054  exexw  2055  spaev  2056  ax12w  2131  nfcriOLD  2896  rspw  3128  reldisj  4382  ralidmw  4435  dtru  5288  bj-ssblem1  34762  bj-ax12w  34785